Linear Elasticity Theory Research Articles

The Effect of Nonlocal Scale Value and Phase Lags on Thermoelastic Waves in a Multilayered LEMV/CFRP Composite Cylinder

In this study, the effect of nonlocal scale value and two phase lags on the free vibration of generalized thermoelastic multilayered LEMV (Linear Elastic Material with Voids)/CFRP (Carbon Fiber Reinforced Polymer) composite cylinder is studied using nonlocal form of linear theory of elasticity. The governing equation of motion is established in longitudinal axis and variable separation model is used to transform the governing equations into a system of differential equations. To investigate vibration analysis from frequency equations, the stress free boundary conditions are adopted at the inner, outer and interface boundaries. The graphical representation of the numerically calculated results for frequency shift, natural frequency, and thermoelastic damping is presented. A special care has been taken to inspect the effect of nonlocal parameter on the aforementioned quantities. The results suggest that the nonlocal scale and the phase lag parameters alter the vibration characteristics of composite cylinders significantly.

Strong unique continuation and global regularity estimates for nanoplates

In this paper, we analyze some properties of a sixth-order elliptic operator arising in the framework of the strain gradient linear elasticity theory for nanoplates in flexural deformation. We first rigorously deduce the weak formulation of the underlying Neumann problem as well as its well posedness. Under some suitable smoothness assumptions on the coefficients and on the geometry, we derive interior and boundary regularity estimates for the solution of the Neumann problem. Finally, for the case of isotropic materials, we obtain new Strong Unique Continuation results in the interior, in the form of doubling inequality and three spheres inequality by a Carleman estimates approach.

Geometrically nonlinear shape sensing of anisotropic composite beam structure using iFEM algorithm and third-order shear deformation theory

The inverse finite element method (iFEM) has been used to achieve the shape sensing of small displacements based on linear elastic theory. However, with the development of smart structure technology, the formulation is not suitable for the anisotropic composite structures with large deformation in practical engineering. Therefore, a nonlinear iFEM algorithm is proposed to monitor the linear and nonlinear deformation of the anisotropic composite beams. The formulation not only involves the effect of orientation of composite fibers on strain distribution into the shape sensing model, but also accounts for the effect of shear deformation without any requirement of shear correction factor. Initially, the third-order shear deformation theory (TSDT) is reviewed along with deriving the nonlinear strain field based on von-Karman strain theory. Considering the problem that the couple term of the shear strain and bending strain is unmeasurable, the relationship between shear and bending displacements is established according to the derived constitutive equations. Then, the proposed nonlinear iFEM method reconstructs the deformed structural shape, where isogeometric analysis (IGA) approach is used to construct the displacement functions and the experimental section strains are calculated from the discretized surface strains. Finally, several examples are solved to verify the proposed methodology. Numerical results demonstrate that the nonlinear iFEM algorithm can improve the reconstruction accuracy by 4% with respect to the linear iFEM method for beam structures. Hence, the proposed approach can be used as a viable tool to predict nonlinear deformation of composite structure.

Boron substitution induced FCC Fe/Cr23C6 interfacial strengthening: An ab initio study

Boron is added to stainless steels and polycrystalline superalloys to promote ductility, corrosion resistance, and retard creep at high temperatures (above 700 °C). However, the role of local atomic environment(s) and how boron may impact the interfacial properties between the Cr23C6 precipitate, and the matrix are not fully understood. In the present work, we consider the face-centered cubic (FCC) Fe (001)/Cr23C6 (001) interface as a model system to study how the interfacial local atomic structure and energy are affected by boron doping using first-principles density functional theory (DFT) calculations. The atomic structure(s) and energetics of a coherent interface were calculated by including the elastic contribution using linear elasticity theory combined with DFT. Boron has a significant effect on the interfacial energy and the atomic arrangement near the interface region through the increased covalent bonding, thereby ordering the interface and making any diffusion processes less favorable energetically. In turn, this may contribute to the retardation of the Cr23C6 precipitate growth and coarsening. Because of the concurrent precipitation of the NbC, Cr23C6, and Sigma phases, it affects the whole cascade of microstructural evolution processes contributing to creep. More specifically, the ancillary additions of boron to the A-terminated Cr23C6 significantly reduce the interfacial energy (∼0.09 J/m2) between the iron matrix and the Cr23C6: from ∼ 0.38 J/m2 without doping to 0.29 J/m2 with doping. This interfacial energy decrease due to boron additions is an important factor in interface strengthening and retarding creep in austenitic stainless steels. Such B-induced short-range order interface strengthening is facilitated via promoting interface covalence bonding and considerable local lattice distortion due to the atomic size misfit. These results were obtained using the new interface model described in the paper. This new interface model allows to quantitatively distinguish the highly complex asymmetric interface structures and corresponding interfacial energies and surface energies.

Radiation force of compressional plane waves on a sphere embedded in an elastic medium

The aim of this work is to derive exact partial-wave series expressions for the radiation force of plane compressional progressive waves propagating inside an elastic medium and incident upon an embedded elastic sphere. The analytical modeling is needed to provide fundamental physical understanding of the underlying phenomenon of mode conversion and its contribution to the acousto-elastic radiation force, and in experimental design. In the context of linear elasticity theory, a rigorous derivation for the acousto-elastic radiation force, based on the integration of the time-averaged radial component of the elastodynamic Poynting vector (or power flow density), is presented and discussed. Initially, the elastic scattering problem is determined and subsequently used to derive the mathematical expression for the acousto-elastic radiation force of progressive compressional waves. The method is also verified using the extended optical theorem for elastic compressional plane waves. Extension to the case of elastic plane standing wave is also provided. Particular importance is made on the contributions of elastic mode preservation (P → P) and mode conversion (P → S) to the acousto-elastic radiation force. Numerical computations for the dimensionless radiation force efficiency and its components demonstrate the importance of compressional-to-shear mode conversion in the scattering by the sphere encased in a linearly-elastic medium. The analytical formalism presented here can be used to validate numerical methods, and the results of the simulations can be utilized as a priori knowledge in optimizing and designing acousto-elastic radiation force experiments involving elastic compressional progressive waves on a sphere in acoustically-engineered materials applications, elasticity imaging methods, activation of implantable devices, characterization of biological tissue, and non-destructive evaluation to name some examples.

Analysis of Stress-Strain State of a Cylinder with Variable Elasticity Moduli Based on Three-Dimensional Equations of Elasticity Theory

Introduction. Functionally graded materials are of great use, because heterogeneity of properties enables to control the strength and rigidity of structures. This has caused great interest in the topic in the world scientific literature. The construction of solutions to such problems depends significantly on the type of boundary conditions. In this paper, we consider the equilibrium of a thin-walled circular cylinder whose mechanical properties change along the radius. Homogeneous boundary conditions were set on cylindrical surfaces that had not been considered before, the effect was on the ends. The mathematical formulation of the problem was carried out in the linear theory of elasticity in the framework of axisymmetric deformation. Expressions were constructed for the components of the stress-strain state of the cylinder, in which some coefficients were found from the solution to the resulting system of linear algebraic equations.Materials and Methods. The material of the cylinder was linearly elastic, the elastic modulus of which depended linearly on the radial coordinate. The basic research method was the asymptotic method, in which half the logarithm of the ratio of the outer and inner radii acted as a small parameter. Iterative processes were used to construct the characteristics of the stress-strain state of the cylinder.Results. Homogeneous solutions to the boundary value problem were obtained for a linearly elastic functionally gradient hollow thin-walled cylinder. An analysis of these solutions made it possible to reveal the nature of the stress-strain state in the cylinder wall. For this purpose, an asymptotic analysis of the solutions was carried out, relations for displacements and stresses were obtained. It was determined that those solutions corresponded to the boundary layer, while their first terms determined Saint-Venant edge effect similar to the plate theory.Discussion and Conclusion. The analytical solution to the equilibrium problem of a thin-walled cylinder inhomogeneous in radius constructed by asymptotic expansion can be used for numerical solution to a specific problem. For this, it is required to solve the obtained systems of linear algebraic equations and determine the corresponding coefficients. The resulting asymptotic representations provide analyzing the three-dimensional stress-strain state. The selection of the number of expansion terms makes it possible to calculate displacements and stresses with a given degree of accuracy. This analysis can be useful in assessing the adequacy of applied calculation methods used in engineering practice.

Effect of Nonlocal Elasticity and Phase Lags on the Magneto Thermoelastic Waves in a Composite Cylinder with Hall Current

In this study, the effect of nonlocal scale value and two phases lags on the free vibration of generalized magneto thermoelastic multilayered LEMV (Linear Elastic Material with Voids)/CFRP (Carbon Fiber Reinforced Polymer) composite cylinder is studied using nonlocal form of linear theory of elasticity. The governing equation of motion is established in longitudinal axis and variable separation model is used to transform the governing equations into a system of differential equations. To investigate vibration analysis from frequency equations, the stress free boundary conditions are adopted at the inner, outer and interface boundaries. The graphical representation of the numerically calculated results for frequency shift, natural frequency, and thermoelastic damping is presented. A special care has been taken to inspect the effect of nonlocal parameter on the aforementioned quantities. The results suggest that the nonlocal scale and the phase lag parameters alter the vibration characteristics of composite cylinders significantly.

Fatigue fracture mechanics in gold-based MEMS notched specimens: experimental and numerical study

The characterization of fatigue fracture mechanics in gold-MEMS notched specimens is presented in this work. A test microstructure with a central notched specimen is specifically designed and built to perform on-chip fatigue test. The central specimen undergoes cyclic loading due to the application of alternating voltage. The variation in the microstructure deflection is measured using an optical profilometer and is attributed to the crack growth in the gold material, causing the variation in the specimen stiffness. The occurrence of pull-in condition is used as a fracture detector, then the fracture of the specimen can be recognized without performing scanning electron microscope inspections during the fatigue test. Crack propagation in the test specimen is simulated through a coupled-field electromechanical fracture finite element model and the resulting crack path is compared to the experimental measurments performed with scanning electron microscope analyses. Finally, Paris’ law is applied and the number of cycles to failure is computed by exploiting the results of the fracture model and experimental measurements. Both experimental and numerical results demonstrate that the notch acts as a stress and strain raiser, fostering crack nucleation, and that the linear elastic fracture mechanics theory is still valid to describe crack propagation in micro-size samples.

Elastohydrodynamic lubrication of soft-layered rollers and tensioned webs in roll-to-plate nanoimprinting

This work presents the development of a numerical model for the elastohydrodynamic lubrication of roll-to-plate nanoimprinting with flexible stamps. Roll-to-plate nanoimprinting is a manufacturing method to replicate micro- and nanotextures on large-area substrates with ultraviolet-curable resins. The roller is equipped with a relatively soft elastomeric layer, which elastically deforms during the imprint process. The elastic deformation is described by linear elasticity theory. It is coupled to the pressure build-up in the liquid resin film, which is described by lubrication theory. The flexible stamp, which is treated as a tensioned web, is pre-tensioned around the roller. The elastic deformation of the tensioned web is described by the large-deflection bending of thin plates equations, considering its non-negligible bending stiffness. A Fischer–Burmeister complementarity condition captures the contact mechanics between the tensioned web and the roller. The governing equations combine in a coupled elastohydrodynamic lubrication model, which is fully described by a set of non-dimensional numbers. These are used in a parameter study to investigate the effect on the pressure and film height distributions. It is shown that the bending stiffness of the tensioned web results in an additional hydrodynamic pressure peak and corresponding minimum in the film height, near the inlet of the roller contact. An increase of the bending stiffness corresponds to a decrease in film height. The numerical results are compared with benchmarks from literature and experimentally validated with layer height measurements from flat layer imprints. Good agreement is found between the numerical and experimental results.

Load response and fatigue life of cement-stabilized macadam base structure considering dynamic and static load differences and tension-compression modulus differences

The conventional semi-rigid base asphalt pavement (SRBAP) construction is designed with the classical theory of linear elastic mechanics. There is a significant discrepancy between the calculation results of the pavement structure load response and the measured results, which leads to inaccurate structural resistance and load response calculations. For this reason, this work depended on the actual structure of the SRBAP, took the linear elasticity theory as a contrast, and considered the difference between dynamic and static loads or the difference between the compressive and tensile modulus of pavement materials. Thus, the pavement structure response to load under three conditions is calculated, and five mechanical indexes are selected for comparative analysis. Furthermore, based on the calculated mechanical response, the structural fatigue life of the cement-stabilized macadam (CSMB) was then calculated, replacing the design index in the fatigue cracking model with the underside tensile stress or equivalent stress. The results demonstrated 20–30% variations between the calculated mechanical responses to dynamic and static stresses. Additionally, there is a substantial discrepancy between the strain computed using the bimodulus theory and the traditional linear elasticity theory, with a maximum divergence of 40%. In the meantime, the fatigue life of the CSMB produced using various design indices is noticeably different, and the fatigue life achieved using equivalent stress as the design index is significantly higher than that obtained using underside course tensile stress. Following is a relationship between the CSMB's fatigue life calculated by underside course tensile stress: viscoelastic theory (dynamic load) > bimodulus theory (tension–compression modulus difference) > classical linear elastic theory; The relationship of fatigue life represented by equivalent stress is as follows: viscoelastic theory > classical linear elastic theory > bimodulus theory.

Computer-aided investigation of the tensile behavior of concrete: Relationship between direct and splitting tensile strength

Currently, a standardised approach for testing the direct tensile strength (fdt) of concrete has yet to be established. Consequently, the correlation between the fdt and the splitting tensile strength (fst) of concrete holds significant importance in evaluating its uniaxial tensile strength. We first review the theoretical relationship between fdt and fst of concrete, then propose a 3D random mesoscale modelling approach to simulate its mechanical behavior under direct and splitting tensile stresses, covering various concrete strength classes (C30-C50). Finally, we derive empirical formulas for the tensile strength and uniaxial compressive strength (fc) via regression analysis of available test data and simulation results, including fst vs. fc, fdt vs. fc, and fdt vs. fst. The findings suggest that the proposed 3D mesoscale model, which takes into account the random characteristics of aggregates such as their shape and distribution, is highly feasible for simulating the tensile behavior of concrete. Widening the load-bearing strip has been shown to enhance the splitting tensile strength, and both fdt and fst can be improved as the concrete strength grade increases. While linear elastic theory implies that the theoretical fdt surpasses fst, the empirical results indicate that the fdt is actually relatively smaller, ranging from 10% to 60% lower than the fst. Moreover, based on the synthesized test and numerical results, a significant power regression relationship has been established between the fst and fdt of concrete.

Effect of double crack on fatigue crack growth life of 3D printing compressor impeller

A compressor impeller with pressure ratio less than 3 is taken as the research object. The 3D printing residual stress was introduced into damage tolerance assessment of blades, and the effect of double crack on Fatigue Crack Growth Life (FCGL) was investigated. The impeller was printed by selective laser melting technology and then heat treated. Finite element method was applied to numerically simulate 3D printing process, heat treatment process and actual working process of impeller, and mechanical quantities such as stress and deformation were completely analyzed. Numerical simulation results agreed well with the experimentally observed crack location in the middle of the blade. Forman–Newman–de Koning model and linear elastic fracture mechanics theory were applied to give the growth law of impeller under single and double crack conditions. The result shows that crack initiation locations of impeller during printing and operating are respectively middle part and root part of blade. The stress intensity factor at the initial crack tips of single and double cracks exhibits a quadratic relationship with rotational speed when only external load is considered. The crack growth paths of single crack in middle and root are radial and axial respectively, and the FCGL of single crack in root is lower than that of single crack in middle. With the increase of rotational speed, the critical crack length decreases linearly and the cycles decreases nonlinearly in both single and double crack conditions. The FCGL of double crack is between two single cracks at low speed and lower than two single cracks at high speed. The critical crack length of double crack is lower than that of corresponding single crack. At 60000rpm, the dominant crack changes from middle crack to root crack.

Carrera Unified Formulation (CUF) for the composite shells of revolution. Equivalent single layer models

Here, higher order models of elastic composite multilayer shells of revolution are developed using the variational principle of virtual power for the 3-D linear anisotropic theory of elasticity and generalized series in the shell thickness coordinates. Following the Unified Carrera Formula (CUF), the stress and strain tensors, as well as the displacement vector, were expanded into series in terms of the coordinates of the shell thickness. The higher-order cylindrical shell supported on the edges under axisymmetric loading, is considered and solved analytically using a Navier close form solution method. Also, composite axisymmetric circular plated as well as parabolic, hyperbolic and pseudo-spheric shell fixed ate the ends are considered. Numerical calculations were performed using the computer algebra software Mathematica. The resulting equations can be used for theoretical analysis and calculation of the stress-strain state, as well as for modeling thin-walled structures used in science, engineering, and technology. The results of calculation can be used as benchmark examples for finite element analysis of the higher order elastic shells.

The Influence of GPL Reinforcements on the Post-Buckling Behavior of FG Porous Rings Subjected to an External Pressure

The work focuses on the post- buckling behavior of functionally graded graphene platelet (FG-GPL)-reinforced porous thick rings with open-cell internal cavities under a uniform external pressure. The generalized rule of mixture and the modified Halpin–Tsai model are here used to evaluate the effective mechanical properties of the ring. Three types of porosity patterns are assumed together with five different GPL distributions as reinforcement across the ring thickness. The theoretical formulation relies on a 2D-plane stress linear elasticity theory and Green strain field in conjunction a virtual work principle to derive the nonlinear governing equations of the post-buckling problem. Unlike the simple ring models, 2D elasticity considers the thickness stretching. The finite element model combined with an iterative Newton–Raphson algorithm is used to obtain the post-buckling path of the ring up to the collapse. A systematic investigation evaluates the effect of the weight fraction of nanofillers, the coefficient of porosity, porosity distribution, and the GPLs distribution on the deep post-buckling path of the ring. Based on the results, it is found that the buckling value and post-buckling strength increase considerably (by approximately 80%) by increasing the weight fraction of the nanofiller of about 1%.

Numerical Analysis on Fatigue Crack Growth at Negative and Positive Stress Ratios.

The finite element method was used to investigate the effect of the stress ratio on fatigue crack propagation behavior within the framework of the linear elastic fracture mechanics theory. The numerical analysis was carried out using ANSYS Mechanical R19.2 with the unstructured mesh method-based separating, morphing, and adaptive remeshing technologies (SMART). Mixed mode fatigue simulations were performed on a modified four-point bending specimen with a non-central hole. A diverse set of stress ratios (R = 0.1, 0.2, 0.3, 0.4, 0.5, -0.1, -0.2, -0.3, -0.4, -0.5), including positive and negative values, is employed to examine the influence of the load ratio on the behavior of the fatigue crack propagation, with particular emphasis on negative R loadings that involve compressive excursions. A consistent decrease in the value of the equivalent stress intensity factor (ΔKeq) is observed as the stress ratio increases. The observation was made that the stress ratio significantly affects both the fatigue life and the distribution of von Mises stress. The results demonstrated a significant correlation between von Mises stress, ΔKeq, and fatigue life cycles. With an increase in the stress ratio, there was a significant decrease in the von Mises stress, accompanied by a rapid increase in the number of fatigue life cycles. The results obtained in this study have been validated by previously published literature on crack growth experiments and numerical simulations.

Torsional buckling response of FG porous thick truncated conical shell panels reinforced by GPLs supporting on Winkler elastic foundation

In the present research, torsional buckling response of thick truncated conical shell panels supporting on Winkler-type elastic medium has been surveyed. The shell is constructed from a porous metal material reinforced by graphene platelets (GPLs) in which the porosity and volume weight fraction of nanofillers are graded along the thickness of shell. Three functions for porosity distribution and five patterns of GPLs are examined. To estimate the Young modulus of the shell, Tsai-Halpin micromechanical model, and for its mass density, extended rule of mixture is employed. Linear theory of 3D elasticity in conjunction with the virtual work principle and numerical graded finite element method (FEM) are applied to derive the state of equilibrium in pre-buckling mode. Torsional buckling forces are derived by applying nonlinear Green strains and by deriving geometric stiffness matrix of the system. The effect of various parameters such as coefficient of porosity, various distributions of porosity and different nanofiller patterns, weight fraction of graphene nanofillers, semi vertex angle of cone, span angle, stiffness of elastic medium and various boundary conditions on torsional buckling loads of functionally graded (FG) porous truncated conical panel reinforced by GPLs have been presented. The main purpose of this research is obtaining the best distribution of porosity and GPLs pattern and investigating the influence of adding nano particles on the torsional buckling forces of the shell. Results denote that employing GPL-X pattern in conjunction with porosity distribution 1 provides the maximum value of buckling loads. Besides by adding the nano particles, the amount of buckling loads will increase 100%.

Failure analysis of an aero-engine inter-shaft bearing due to clearance between the outer ring and its housing

The inter-shaft bearing is a critical but vulnerable component of dual rotors, because of the severe operational conditions and loads. This paper investigates the failure mechanism of a fractured inter-shaft bearing belonging to a high thrust-weight ratio turbofan engine. Fractographic and microscopic analyses of the fracture surfaces were conducted, followed by vibration signal processing. Finite element analyses of the outer ring were also performed considering the centrifugal, thermal, and roller-raceway loads. What’s more, using the linear elastic fracture mechanics theory, the crack growth path simulation was obtained, whose result showed good agreement with the real fracture morphologies. The results indicated that due to the clearance between the outer ring and its housing during the operation, the anti-rotation tabs of the outer ring were subjected to non-uniform load, thus several tabs might suffer large stress concentrations at the root fillets, which could lead to the crack initiation. During the following operation of the aero-engine, the crack propagated through the outer ring and cause the fatigue fracture of tabs. The investigation provided herein suggests that the outer ring and its housing should be redesigned to eliminate the clearance during the operation, an outer ring formed integrally with the bearing housing flange is one of the failure preventions of this failure mode.

On the Equilibrium of Nonthin Cylindrical Shells with a Dent

On the basis of the linear three-dimensional theory of elasticity, we consider the problem of equilibrium of nonthin isotropic cylindrical shells with dents under certain boundary conditions imposed on the end faces. To describe the cross section of the reference surface, we use the Pascal snail equation in polar coordinates. By the method of separation of variables, with the help of the approximation of functions by discrete Fourier series, we reduce the three-dimensional boundary-value problem to a one-dimensional problem, which is solved by the stable numerical method of discrete orthogonalization. We estimate the accuracy of the obtained solutions. The results of solving the problems are presented in the form of plots and tables.

Численная реализация метода переменных параметров при решении упругопластических задач на основе графовой модели упругого тела

The article highlights the results of using a graph model of a continuous medium for solving elastic-plastic problems. The stress-strain state is determined by the method of variable parameters based on the deformation diagram of the medium material. The method is based on the representation of the defining elastic plasticity relations in the form of equations of linear elasticity theory, but with variable elasticity parameters. The computational process is an iterative procedure in which each successive approximation is reduced to solving a linear elastic problem. The problem is solved by the graph method. The stress-strain state is found by a non-standard numerical method in which a solid body is represented by a discrete model in the form of an oriented graph. Concrete examples show the high efficiency of the method in comparison with the traditional finite element method. The increased accuracy of calculations, even when using coarse grids, is ensured by the fact that: 1) the vertex and contour laws of graph theory implement the equations of equilibrium and compatibility of deformations for the element as a whole; 2) the equilibrium equations are performed locally by the volume of the element. The problems of the elastic-plastic bending of the console and the elastic-plastic state of a plate with a circular hole are solved as examples. Comparison of the obtained results with the solutions of these problems by other methods showed a good match. The high accuracy of calculations makes it possible to use the iterative procedure of the method of variable elasticity parameters as a subroutine in the application package created for the graph method.

Reconstructing Loads in Nanoplates from Dynamic Data

It was recently proved that the knowledge of the transverse displacement of a nanoplate in an open subset of its mid-plane, measured for any interval of time, allows for the unique determination of the spatial components {fm(x,y)}m=1M of the transverse load ∑m=1Mgm(t)fm(x,y), where M≥1 and {gm(t)}m=1M is a known set of linearly independent functions of the time variable. The nanoplate mechanical model is built within the strain gradient linear elasticity theory, according to the Kirchhoff–Love kinematic assumptions. In this paper, we derive a reconstruction algorithm for the above inverse source problem, and we implement a numerical procedure based on a finite element spatial discretization to approximate the loads {fm(x,y)}m=1M. The computations are developed for a uniform rectangular nanoplate clamped at the boundary. The sensitivity of the results with respect to the main parameters that influence the identification is analyzed in detail. The adoption of a regularization scheme based on the singular value decomposition turns out to be decisive for the accuracy and stability of the reconstruction.